1. Dedication:

This Project is dedicated to my two daughters. I love you both equally, from the furthest Sun our Universe will ever know and back! Love, Dad.

2. Installation:

Note: Due to known loading issues, load the following package(s) dependency(ies), if needed ¹

install.packages("hms", dependencies = TRUE)
install.packages("readr", dependencies=TRUE, 
                 INSTALL_opts = c('--no-lock'))

Load the following Library(ies)

library(tidyverse)
library(caret)
library(data.table)
library(RColorBrewer)
library(rmarkdown)
library(dslabs)
library(gtable)
library(hexbin)
library(hms)
library(readr)
library(gt)
library(dplyr)
library(ggpmisc)
library(gridExtra)
library(janitor)
library(lubridate)
library(highcharter)
library(viridisLite)
library(broom)
library(scales)
library(xfun)
library(htmltools)
library(mime)
library(quantmod)
library(forecast)
library(tseries)
library(ggfortify)
library(png)
library(jpeg)
library(gtsummary)
library(latexpdf)
library(tinytex)
library(ggforce)

3. Reference Section:

Bond Yields and Interest Rates: 1900 to 2002. (2003). US CENSUS. Retrieved August 18, 2022, from (https://www2.census.gov/library/publications/2004/compendia/statab/ 123ed/hist/hs-39.pdf) The U.S. Census tracked The 3 Month Bond Yield from 1900 to 2002. The 3 Month Bond Yield is closely correlated with Federal Funds Rate. I used the 3 Month Bond Yield to fill in missing Federal Funds Rate data from 1900-1951.
Amadeo, K. (2022, July 27). US Inflation Rate by Year: 1929–2023. The Balance. Retrieved August 18, 2022, from (Https://www.thebalance.com/u-s-inflation-rate-history-by-year-and-forecast-3306093)
Irizarry, R. A. (2022, July 7). Introduction to Data Science. HARVARD Data Science. Retrieved August 8, 2022, from (Https://rafalab.github.io/dsbook/)
Wheelock, D. C. (2021, September 13). Overview: The History of the Federal Reserve. Federal Reserve History. Retrieved August 8, 2022, from (https://www.federalreservehistory.org/essays/federal-reserve-history)
Julian G.F. (2022, May 10). U.S Inflation - Analysis in R. Kaggle. Retrieved August 8, 2022, from (https://www.kaggle.com/code/fit4kz/u-s-inflation-analysis-in-r)
Standard and Poor’s (S&P) 500 Index Data including Dividend, Earnings and P/E Ratio. (n.d.). DataHub. Retrieved August 8, 2022, from (https://datahub.io/core/s-and-p-500)
Bloomberg. (2022, August 15). Inside the Founding of the Federal Reserve [Video]. YouTube. (https://www.youtube.com/watch?v=0hzdglWpxVM&t=314s) Author and journalist Roger Lowenstein describes the economic crises that led to the founding of the US Federal Reserve in 1913.
U.S. Bureau of Labor Statistics. (2022). CPI Home : U.S. Bureau of Labor Statistics. (https://www.bls.gov/cpi/)
Standard and Poor’s 500 (S&P 500) - Explained. (n.d.). The Business Professor, LLC. Retrieved August 22, 2022, from (https://thebusinessprofessor.com/en_US/investments-trading-financial-markets/standard-and-poors-500-sp-500-definition)
Introduction to ARIMA models. (2019). Duke.edu. (https://people.duke.edu/~rnau/411arim.htm)
Long, J. (2019, September 26). 14 Time Series Analysis | R Cookbook, 2nd Edition. Retrieved September 5, 2022, from (https://rc2e.com/timeseriesanalysis)
Srivastav, A. K. (2022, September 13). Pearson correlation coefficient. WallStreetMojo. Retrieved September 18, 2022, from https://www.wallstreetmojo.com/pearson-correlation-coefficient/

4. Overview:

Goal One: To examine the data to identify any correlation using Pearson’s Correlation Coefficient (r).

Goal Two: Create a forecasting machine learning model using past data from 1929-2017 to predict inflation and the appropriate federal funds rate.

Since 1929, the U.S. has combated inflation. An inflation rate of 2% is believed to be an excellent environment for businesses and consumers. During deflation, corporations and local businesses lose pricing power. Businesses have to shed employees, future investments, and goods to maintain a profit which causes an economic slowdown during deflationary periods. During rising inflation above 2%, business profits rise temporally, but consumer pricing power is eroded over time, and it can lead to hyperinflation/economic crisis/economic slowdown.

To prevent reoccurring economic collapses, deflation, galloping inflation and to fix the lack of synergy with the other 12 regional banks, The U.S. founded the Federal Reserve (the central bank) on December 23, 1913. In this project, I will explore if correlations exist within the Monthly U.S. Consumer Price Index (CPI) average for all U.S. cities, Inflation Rate Year over Year (YoY), geopolitical events, economic events, GDP growth, Federal Funds Rate, and S&P 500 price annualized from 1929 to 2017. I will also examine if one of the Federal Reserve most powerful tools, the Federal Funds Rate, is correlated with several factors listed above. I will create a forecasting algorithm using back dated information to predict inflation and appropriate federal fund rate to combat inflation.

To examine if the United States’ geopolitical, domestic, and economic events are correlated with Inflation Rate YoY. I will also examine how the Federal Reserve Fund Rate affects the following: Monthly U.S. Consumer Price Index (CPI) average for all U.S. cities, Inflation Rate YoY, Geopolitical events, Economic Events, GDP Growth, and S&P 500 annualized prices utilizing the Pearson’s Correlation Coefficient (r). I will also create a forecasting machine learning model using back dated information to predict inflation and appropriate federal funds rate

5. Data Wrangling: Clean! Clean! Clean!:

Goal: Clean the data. Display the Federal Reserve Fund Rate, the Monthly U.S. Consumer Price Index (CPI) average for all U.S. cities, Inflation Year over Year (YoY), Geopolitical events, Economic Events, G.D.P. Growth, and S&P 500 price annualized from 1929-2017.

The Standard & Poor’s earliest origins can be linked to the stock market in 1923. The Standard & Poor’s index at the time contained 233 companies. Today, it has 500 companies within its index. It is widely tracked by economists, politicians, investors, and speculators. It is often considered an early indicator of a possible economic expansion or slowdown.

5.1 Review S&P 500 Data

Annual Closing Price	Calendar_Year
S&P 500 Annualized Closing Price
HardvardX Capstone Project 2022
$26.02	1929
$21.03	1930
$13.66	1931
$6.93	1932
$8.96	1933
$9.84	1934
$10.60	1935
$15.47	1936
$15.41	1937
$11.49	1938
Portions of this data is from the Reference Section.
S&P 500 Data is from 1929-2017

5.2 Review CPI Data

Per the U.S. Bureau of Labor Statistics (“U.S. Bureau of Labor Statistics”, 2002), The Consumer Price Index (CPI) is the most widely used measure of inflation and is an indicator of the effectiveness of government policy. CPI is calculated by recording the group of goods, services, and housing that urban consumers purchase and the price average change on a monthly basis.

Annual_CPI_Average	Calendar_Year
Average U.S. CPI Accumulated Data Annualized
HardvardX Capstone Project 2022
17.1583%	1929
16.7000%	1930
15.2083%	1931
13.6417%	1932
12.9333%	1933
13.3833%	1934
13.7250%	1935
13.8667%	1936
14.3833%	1937
14.0917%	1938
Portions of this data is from the Reference Section
CPI Data is based on data from 1929-2017

5.3 Review U.S. Economic and Event data

Since 1929, The United States has experienced different economic and geopolitical events. The Federal Reserve monitors said events and create policies to accommodate the economy to prevent another Great Depression scenario. Federal Reserve utilizes its “set of tools” to help promote a healthy business cycle based on their mandates.

A Business Cycle is the beginning of an expansion(post-recession / post-economic slowdown) period and the beginning of a contraction period (recession/economic slowdown).

The Inflation Rate YoY is the rate of change of inflation yearly. The Inflation rate YoY differs from CPI Annualized data. CPI Annualized data shows how the value of products in 1929 appreciates every year until 2017 based on average inflation accumulated each year. For instance, a gallon of milk in Hawaii on the island of Oahu cost 26 cents in 1929 now a gallon of milk on the island of Oahu cost $5.50. Thats a whooping 2115% of accumulated inflation exceeding the 2017 percentage by 10x. The Inflation Rate YoY shows the change in annual inflation in each specific year vice compounding year after year. The table below will show the changes for this metric.

GDP is the total of all goods produced and sold by a nation over a specific period. This is an indicator of economic growth, stagnation, or slowing down. Let us take a look at U.S. Economic data, Geopolitical Events, and Federal Reserve Data.

Year	Inflation Rate YoY	Fed Funds Rate	Business Cycle	GDP Growth	Events Affecting Inflation
U.S. Economic, Geopolitical Events and Federal Reserve Data
HardvardX Capstone Project 2022
1929	0.60%	4.42%	August peak	6.52%	Market crash
1930	-6.40%	2.23%	Contraction	-8.50%	Smoot-Hawley
1931	-9.30%	1.40%	Contraction	-6.40%	Dust Bowl
1932	-10.30%	0.88%	Contraction	-12.90%	Hoover tax hikes
1933	0.80%	0.52%	Contraction ended in March	-1.20%	FDR's New Deal
1934	1.50%	0.26%	Expansion	10.80%	U.S. debt rose
1935	3.00%	0.14%	Expansion	8.90%	Social Security
1936	1.40%	0.14%	Expansion	12.90%	FDR tax hikes
1937	2.90%	0.45%	Expansion peaked in May	5.10%	Depression resumes
1938	-2.80%	0.05%	Contraction ended in June	-3.30%	Depression ended
Portions of this data within this table is from Reference Section
Federal Funds Rate is based on data from The 3 Month Bond Yield from 1929-1954

Now that we have a better look at the data, it is hard to discern which economic event, geopolitical event or federal reserve action data correlates with one another. Let us visualize the data to see if we can find an inverse, positive, or no correlation.

6. Data Visualization: Plotting the Cleaned Data:

Goal: Create visualizations with individual data and combined data. Observe any inverse, none, or positive correlations.

6.1. Chart of CPI Annualized Data from 1929 to 2017

6.2. Chart of S&P 500 from 1929 to 2017

6.3. Chart of Comparing Average CPI Annualized and S&P 500

The chart above shows how the Consumer Price Index has grown exponentially over time with the S&P 500. The geopolitical and economic events reflect the S&P 500 negative/positive reactions in some cases and nil in others. As CPI has grew gradually from the late 1970s, the S&P 500 has continued to grow faster in worth over time.

6.4. Chart of Federal Funds Rate Annualized

Federal Reserve’s Fed Funds Rate is a tool utilized by the Federal Reserve to tackle inflation, economic slowdowns or promote growth in the economy. Chart of Federal Funds Rate from 1929 - 2017.

6.5. Chart of Federal Funds Rate and Inflation Rate YoY

Inflation at high levels is one of the most significant issues that can cause an economic slowdown. Chart of Federal Reserve’s Fed Funds Rate and Inflation Rate YoY.

Looking at the chart above, we can assess that the Federal Funds Rate and Inflation Rate YoY tend to trend in the same direction annually (the data overlap). We will dig deeper into the data later for confirmation.

6.6. Chart of GDP Growth Annualized

Negative GDP signals economic slowdown, a neutral rate indicates economic stagnation, and a positive rising GDP rate signals economic expansion. Let’s take a look at GDP. Chart of GDP Growth with the Average GDP.

6.7. Chart of GDP Growth and Inflation Rate YoY

After looking at the visualizations, we noticed that some data might have a positive correlation while others have an inverse or no correlation.

I also noticed that U.S. CPI and S&P 500 annualized are more exponential growth over time than the other variables. The Federal Reserve is not mandated to manage the S&P 500 and is banned from buying stocks per the Federal Reserve Act. For this purpose, we will only examine Inflation YoY, GDP Growth, and CPI Average Annualized versus the Federal Reserve’s Fed Funds Rate. We will use Pearson’s Correlation Coefficient in our Data Analysis - Correlation Section to accurately compute the correlations.

7. Data Analysis: Discovering Correlations

Goal: To observe if the Fed Funds Rate has a positive, negative or no correlation with Inflation YoY, GDP Growth and CPI Average Annualized from 1929 - 2017.

I will use Pearson’s Correlation Coefficient. Pearson’s Correlation Coefficient measures the linear correlation between two variables. For the Pearson’s Correlation Coefficient, the value “r” represents the correlation.

If r = 0:1, this means an absolute correlation (the variables move in the same direction). If r = 0, this means no correlation between the two variables, and the value r = 0:-1 means a negative correlation (the variables move in the inverse direction). For more information on Pearson’s Correlation or any correlation formula, please refer to the Reference Section.

Pearson Correlation Coefficient Formula (Srivastav, 2022): r =

\[{n}(\sum xy)- (\sum x) (\sum y)\] \[\\\sqrt{[n\sum x^2 - (\sum x)^2] [n\sum y^2 - (\sum y)^2]}\]

r = correlation coefficient. n = number of pairs of scores. $x$ = values of the x-variable in a sample.
$y$ = values of the y-variable in a sample.
$\sum$ = sum of.

Year	Inflation Rate YoY	Fed Funds Rate	GDP Growth	Annual CPI Average
Variables for Pearson's (r) Data Examination
HardvardX Capstone Project 2022
1929	0.6	4.42	6.52	17.1583
1930	-6.4	2.23	-8.50	16.7000
1931	-9.3	1.40	-6.40	15.2083
1932	-10.3	0.88	-12.90	13.6417
1933	0.8	0.52	-1.20	12.9333
1934	1.5	0.26	10.80	13.3833
1935	3.0	0.14	8.90	13.7250
1936	1.4	0.14	12.90	13.8667
1937	2.9	0.45	5.10	14.3833
1938	-2.8	0.05	-3.30	14.0917
Portions of this data is from the Reference Section.
All Data is from 1929-2017

7.1 Federal Funds Rate vs U.S. Inflation Rate YoY

## 
##  Pearson's product-moment correlation
## 
## data:  All_CorrData$"Fed Funds Rate" and All_CorrData$"Inflation Rate YoY"
## t = 4.9002, df = 87, p-value = 4.392e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2843698 0.6138815
## sample estimates:
##       cor 
## 0.4650834

Since the correlation is .4650834, we can see we have a positive but moderate correlation with Federal Funds Rate and Inflation Rate YoY

To compute the amount of variation between each variable we will utilize R² and convert it to a percentage. With a R² of 22% this means that 78% of variance is explained by unknown factors.

7.2 Federal Funds Rate vs GDP Growth

cor.test(All_CorrData$'Fed Funds Rate', All_CorrData$'GDP Growth')

## 
##  Pearson's product-moment correlation
## 
## data:  All_CorrData$"Fed Funds Rate" and All_CorrData$"GDP Growth"
## t = -0.29098, df = 87, p-value = 0.7718
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.2378934  0.1782326
## sample estimates:
##        cor 
## -0.0311815

Since the correlation is -0.0311815, we can see we have a negative to no correlation with Federal Funds Rate and GDP Growth

To compute the amount of variation between each variable we will utilize R² and convert it to a percentage (remove the negative number for r will not compute). With a R² of 0% this means that 100% of variance is explained by unknown factors.

7.3 Federal Funds Rate vs Annual CPI Average.

cor.test(All_CorrData$'Fed Funds Rate', 
         All_CorrData$'Annual CPI Average')

## 
##  Pearson's product-moment correlation
## 
## data:  All_CorrData$"Fed Funds Rate" and All_CorrData$"Annual CPI Average"
## t = 0.23721, df = 87, p-value = 0.8131
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.1838068  0.2324491
## sample estimates:
##        cor 
## 0.02542312

Since the correlation is 0.02542312, we can see we have a positive but little correlation with Federal Funds Rate and Annual CPI Average

To compute the amount of variation between each variable we will utilize R² and convert it to a percentage. With a R² of 0% this means that 100% of variance is explained by unknown factors.

7.4 Summary Table of the Pearson’s Correlation 1929-2017 results

Fed Funds Rate vs Inflation YoY	Fed Funds Rate vs GDP Growth	Fed Funds Rate vs Annual CPI Average
Pearson's (r) Data Variable Variation (%)
HardvardX Capstone Project 2022
22	0	0
Portions of this data is from the Reference Section.
The data is from 1929-2017

As we can see, our best correlation with the Fed Funds Rate is Inflation Rate YoY. An R² of 22% means that unknown factors explain 78% of the variance. The unknown factors could be outliers. Outliers affect the accuracy of a Pearson’s Correlation formula. Let’s create Linear Regression charts to view the correlations and see if we have any outliers.

8. Data Analysis: Correlation Visualization

Goal: To visualize the data. Observe what factor(s) are causing the major divergences in correlation.

8.1 Scatterplot Pearson’s Correlation Coefficient (r) Formula for Federal Funds Rate vs U.S. Inflation Rate YoY

8.2 Linear Regression Chart of the Pearson’s Correlation Coefficient (r) for Fed Funds Rate and Inflation Rate YoY from 1929-2017

8.3 Scatterplot Pearson’s Correlation Coefficient (r) Formula for Federal Funds Rate vs U.S. GDP Growth

8.4 Linear Regression Chart of the Pearson’s Correlation Coefficient (r) for Fed Funds Rate and GDP Growth from 1929-2017.

8.5 Scatterplot Pearson’s Correlation Coefficient (r) Formula for Federal Funds Rate vs CPI Accumulated Average Annualized

8.6 Linear Regression Chart of the Pearson’s Correlation Coefficient (r) for Fed Funds Rate and Annual CPI Average from 1929-2017

8.7 Outliers

As depicted in each chart, outliers can affect Pearson’s correlation accuracy, but we also can see that all other data displayed in each chart had a distinct or indistinct correlation.

On a good note, we did have a positive but moderate correlation with Federal Funds Rate and Inflation Rate YoY. An R² of 22% means that unknown factors explain 78% of the variance.

Let us chart the economic and geopolitical events with Federal Funds Rate and Inflation Rate YoY to see if the “outliers” were economic/geopolitical driven.

In addition let us plot all GDP data points that are greater than or equal to the GDP annualized average of 3.38.

As we can see, outliers are caused by economic and geopolitical factors. These factors can affect inflation and the Federal Reserve’s Fund Rate.

8.8 Federal Funds Rate vs Inflation Rate YoY Outliers

The majority of the outliers happened prior to 1951. Let us look at a chart highlighting outliers and correlated data for the Federal Funds Rate vs Inflation Rate YoY from 1929-2017.

As you can see, before 1951, there were 11 outliers and only six after 1951. This may be attributed to the Federal Reserve and U.S. Treasury signing the Accord in 1951. This Accord allowed the Federal Reserve to act independently, utilize its economic tools to fight inflation, and implement monetary policy.

Lets see if the correlation changes if I remove 1929-1950 data. Test the Pearson’s Correlation Coefficient (r) Formula for Federal Funds Rate vs U.S. Inflation Rate YoY

Since the correlation is 0.7710506, we can see we have a positive but moderate correlation with Federal Funds Rate and Inflation Rate YoY

To compute the amount of variation between each variable we will utilize R² and convert it to a percentage

With a R² of 59% this means that 41% of variance is explained by unknown factors.

Scatterplot Pearson’s Correlation Coefficient (r) Formula for Federal Funds Rate vs U.S. Inflation Rate YoY

Linear Regression Chart of the Pearson’s (r) for Fed Funds Rate and Inflation Rate YoY from 1951-2017

8.9 Pearson’s Correlation Results:

Fed Funds Rate vs Inflation YoY after 1951	Fed Funds Rate vs Inflation YoY before 1950	Fed Funds Rate vs GDP Growth	Fed Funds Rate vs Annual CPI Average
Final Pearson's (r) Variable Variation Data (%)
HardvardX Capstone Project 2022
59	22	0	0
Portions of this data is from the Reference Section.

After focusing on when the Federal Reserve could use the full range of its tools to combat Inflation Rate YoY and removing the outliers from 1929-1950, we achieved a Pearson’s (r) of 0.7710506 and an R² of 59%. This means that unknown factors explain 41% of the variance. We identified the unknown factors in our visualizations. We know with high confidence that the Federal Fund Rate positively correlates with the Inflation Rate YoY.

Ratio Federal Funds Rate vs Inflation Rate YoY to determine Forecasting Model Tolerance. Lets round the tolerance to the nearest whole number for simplicity. We will use this in our forecasting model.

round(Model_Tolerance)

## [1] 1

9. Machine Learning: Forecasting and Backtesting Model for Inflation Rate YoY:

Goal: Create a Machine Learning Inflation Rate Year over Year Forecasting and Backtesting Model.

Using Pearson’s Correlation, we concluded that the Inflation Rate YoY has a strong positive correlation with Federal Reserve’s Federal Funds Rate. Let us create a machine learning forecasting model to predict future Inflation Rate YoY and Federal Funds Rate. To make this model sustainable, we will have to backtest with primary data used above. To conduct this, we will use a time series model. Specifically, our machine learning model will use the AutoRegressive Integrated Moving Average (ARIMA). Remember that outliers can drive inconsistencies in our data model, so we want to ensure that our model is within tolerance for most of our data. For this backtesting and forecasting this model we will use a data prediction tolerance of +/- 1 point or 1000 basis points of the original data

9.1 Inflation Rate YoY Forecasting Model Data

As you can see, the data is scattered about. To use the ARIMA model, we will need to verify the data and see if it is in a time series format. If not, convert it.

9.2 Convert Inflation Rate YoY Forecasting Model Data

9.3 Backtest the Inflation Rate YoY Forecasting Model

Now that we have the data properly formatted, we have to verify if the data is stationary. Per the Duke research team (“Introduction to ARIMA models,” 2019),“A stationary series has no trend, its variations around its mean have a constant amplitude, and it wiggles consistently, i.e., its short-term random time patterns always look the same in a statistical sense. With that being established, the Armia model requires stationary data to predict future values from older data properly.”

In this case, we are using data from 1929-2017. We will predict ten years from 2018-2028 and backtest the data from 1993-2003. First, let us backtest the data to find a suitable model to predict future values. We will do a ten year model from 1993-2003.

## Time Series:
## Start = 1929 
## End = 1992 
## Frequency = 1 
##  [1]   0.6  -6.4  -9.3 -10.3   0.8   1.5   3.0   1.4   2.9  -2.8   0.0   0.7
## [13]   9.9   9.0   3.0   2.3   2.2  18.1   8.8   3.0  -2.1   5.9   6.0   0.8
## [25]   0.7  -0.7   0.4   3.0   2.9   1.8   1.7   1.4   0.7   1.3   1.6   1.0
## [37]   1.9   3.5   3.0   4.7   6.2   5.6   3.3   3.4   8.7  12.3   6.9   4.9
## [49]   6.7   9.0  13.3  12.5   8.9   3.8   3.8   3.9   3.8   1.1   4.4   4.4
## [61]   4.6   6.1   3.1   2.9

ACF shows us similarities over time using lagged data in a time series.
The autocorrelations within the blue upper and lower limits are considered significant. The insignificant autocorrelations exceed the blue upper and lower limits. In this ACF data plot, more data units pass the blue upper line, indicating that the data is not as stationary. Lag is a specific period of time; we will reference the lag with a number, i.e., lag 1.

The PACF data is measured by extracting the effects of any shorter lag correlations. In an ARIMA model, PACF can pinpoint the number of autoregression coefficients. PACF test is another verification that the data is not stationary due to the spikes in data that passed the blue upper and lower lines.

Our final verification will be the Augmented Dickey-Fuller Test. This test will determine whether the model data is stationary or nonstationary. P value if less than .05 means it is statistically significant.

## 
##  Augmented Dickey-Fuller Test
## 
## data:  Inflation_Model_1993_Time
## Dickey-Fuller = -2.9543, Lag order = 3, p-value = 0.1883
## alternative hypothesis: stationary

Our P-Value of .18 (18%) is well above our minimum of .05 (5%). This means we have to alter our confidence interval to 82. The ARIMA model we will use comprises three principles. P is the total amount of autoregressive terms, D is the amount of non-seasonal differences needed for the data to remain stationary, and Q is the amount of lagged forecasting errors in the prediction equation. This format is mirrored in the ARIMA (p,d,q). Selecting the correct ARIMA (p,d,q) is critical for this forecasting model. Null Hypothesis means autocorrelation does not exist, and Alternate Hypothesis means autocorrelation does not exist.

True_Inflation_Model_1993= auto.arima(
  Inflation_Model_1993_Time, ic="aic", trace= TRUE)

## 
##  ARIMA(2,1,2) with drift         : 346.7356
##  ARIMA(0,1,0) with drift         : 359.3801
##  ARIMA(1,1,0) with drift         : 361.0432
##  ARIMA(0,1,1) with drift         : 358.9609
##  ARIMA(0,1,0)                    : 357.3852
##  ARIMA(1,1,2) with drift         : Inf
##  ARIMA(2,1,1) with drift         : 349.6831
##  ARIMA(3,1,2) with drift         : 342.7439
##  ARIMA(3,1,1) with drift         : 340.7826
##  ARIMA(3,1,0) with drift         : 338.9625
##  ARIMA(2,1,0) with drift         : 359.0333
##  ARIMA(4,1,0) with drift         : 340.754
##  ARIMA(4,1,1) with drift         : Inf
##  ARIMA(3,1,0)                    : 337.4522
##  ARIMA(2,1,0)                    : 357.0865
##  ARIMA(4,1,0)                    : 339.2993
##  ARIMA(3,1,1)                    : 339.3246
##  ARIMA(2,1,1)                    : 348.1981
##  ARIMA(4,1,1)                    : Inf
## 
##  Best model: ARIMA(3,1,0)

The best ARIMA for our model will be ARIMA(3,1,0), verify that the data is stationary and smoothed

Now that the data is smoothed and fits our current ARIMA model, let us forecast what inflation would be in 10 years starting from 1993. Note: h = the number of years in the future. Level 82 = the confidence interval for our model.

True_Inflation_Model_1993

## Series: Inflation_Model_1993_Time 
## ARIMA(3,1,0) 
## 
## Coefficients:
##           ar1      ar2      ar3
##       -0.2472  -0.3188  -0.5458
## s.e.   0.1073   0.1035   0.1049
## 
## sigma^2 = 11.26:  log likelihood = -164.73
## AIC=337.45   AICc=338.14   BIC=346.02

Inflation_Model_Forecast_1993 = forecast(
  True_Inflation_Model_1993, level = c(95), h = 11)

Lets validate the data using the Ljung Box.test to verify that the residuals are not just “white noise”

## 
##  Box-Ljung test
## 
## data:  Inflation_Model_Forecast_1993
## X-squared = 0.0073422, df = 1, p-value = 0.9317

If the P value is less than .05 that means that the data has autocorrelation significance with a 95% confidence interval.

## 
##  Box-Ljung test
## 
## data:  Inflation_Model_Forecast_1993
## X-squared = 8.9147, df = 5, p-value = 0.1125

Inflation_Model_1993_2003 = auto.arima(
  Inflation_Model_1993_Time, ic="aic", trace= TRUE,) 

Backtest_Inflation_Model_Forecast = forecast(
  Inflation_Model_1993_2003 , level = c(95), h = 11)

Let’s examine how many years are within +/- 1000 basis points of the original Fed Data object Inflation_Model.

The model has more false results than true. This data is nearly double the inflation rate based on our historical data. We have to make several adjustments moving forward. Let us view the backtest model vs actual data chart.

Let’s adjust the arima(p,d,q), review and update our P-Value.

Inflation_Model_1993_2003 = arima(
  fixed = NULL, Inflation_Model_1993_Time, 
  order = c(6,2,1), transform.pars=TRUE)

Backtest_Inflation_Model_Forecast_1 = forecast(
  Inflation_Model_1993_2003, level = c(82), h = 11)

Verify our model by running check residual and update the conf. This is a series of diagnostic tests that we will use to validate our model (Long, 2019).

checkresiduals(Backtest_Inflation_Model_Forecast_1)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(6,2,1)
## Q* = 5.5324, df = 3, p-value = 0.1367
## 
## Model df: 7.   Total lags used: 10

Backtest_Inflation_Model_Forecast_1 = forecast(
  Inflation_Model_1993_2003, level = c(87), h = 11)

Great improvement.

Per (Long, 2019), to verify check residuals, we should look for the following:

Do the standardized residuals show volatility clusters?
- If so, readjust the model.
Does the ACF show significant autocorrelation between the residuals?
- If so, readjust the model.
Do the residuals look bell-shaped?
- If so, this suggests they are reasonably symmetrical.
Are the p-value in the Ljung–Box test extensive?
- If so, this indicates the residuals are patternless meaning the model has extracted all the information, and the only noise is left behind.

Let us compare the data and adjust the confidence interval to 87% based on the new P-Value of .13 (13%). ²

Year	Inflation_YoY	Model_Inflation_YoY	Predict_Diff	Within_Tolerance
Inflation Rate Year over Year Model
HardvardX Capstone Project 2022
1993	2.7	1.6	1.1	FALSE
1994	2.7	3.7	-1.0	TRUE
1995	2.5	3.7	-1.2	TRUE
1996	3.3	3.2	0.1	TRUE
1997	1.7	2.2	-0.5	TRUE
1998	1.6	1.9	-0.3	TRUE
1999	2.7	2.3	0.4	TRUE
2000	3.4	2.0	1.4	FALSE
2001	1.6	2.0	-0.4	TRUE
2002	2.4	1.5	0.9	TRUE
2003	1.9	1.5	0.4	TRUE
Portions of this data is from the Reference Section
Inflation Rate YoY is based on data from 1993-2003

Amazing!
Our model is now closer to our goal. Let us view it in a chart.

9.4 Create a Future Inflation Rate YoY Forecasting Model

Now that we have a great model we created when we backtested the data, let us create a model that can predict future values of the Inflation Rate YoY. Remember, we must verify the data, utilize the ARIMA from our backtest model, verify that the data is stationary and smoothed.

Now that the data is smoothed and fits our current ARIMA model lets forecast what inflation would be in 10 years starting from 2018 and verify the model.

Inflation_Model_Forecast = forecast(
  True_Inflation_Model, level = c(81), h = 11)

checkresiduals(Inflation_Model_Forecast)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(6,2,1)
## Q* = 6.1637, df = 3, p-value = 0.1039
## 
## Model df: 7.   Total lags used: 10

Lets update the confidence interval to 90% since the P Value is .10 (10%)

Amazing now we have a visual of what Inflation Rate YoY would be in the future based on our inputs into the forecasting model.

10. Machine Learning: Forecasting and Backtesting Model for Fed Funds Rate:

Goal: Create a Machine Learning Federal Reserve Funds Rate Forecasting and Backtest Model.

10.1 Federal Reserve’s Fed Funds Rate Model Data

Let us create a Federal Reserve’s Fed Funds Rate model to predict the future rate and backtest prior data.

10.2 Convert Inflation Rate YoY Forecasting Model Data

10.3 Backtest the Federal Reserve’s Fed Fund Rate Forecasting Model

adf.test(Fed_Model_1993_Time)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  Fed_Model_1993_Time
## Dickey-Fuller = -2.3703, Lag order = 3, p-value = 0.4249
## alternative hypothesis: stationary

Fed_Model_1993_2003 = auto.arima(
  Fed_Model_1993_Time, ic="aic", trace= TRUE,)

## 
##  ARIMA(2,1,2) with drift         : 258.221
##  ARIMA(0,1,0) with drift         : 255.8698
##  ARIMA(1,1,0) with drift         : 257.8299
##  ARIMA(0,1,1) with drift         : 257.806
##  ARIMA(0,1,0)                    : 253.8799
##  ARIMA(1,1,1) with drift         : Inf
## 
##  Best model: ARIMA(0,1,0)

Backtest_Fed_Model_Forecast = forecast(
  Fed_Model_1993_2003, level = c(58), h = 11)

Backtest_Fed_Model_Forecast

##      Point Forecast      Lo 58    Hi 58
## 1993              3  1.5595451 4.440455
## 1994              3  0.9628891 5.037111
## 1995              3  0.5050589 5.494941
## 1996              3  0.1190901 5.880910
## 1997              3 -0.2209552 6.220955
## 1998              3 -0.5283796 6.528380
## 1999              3 -0.8110855 6.811086
## 2000              3 -1.0742218 7.074222
## 2001              3 -1.3213648 7.321365
## 2002              3 -1.5551185 7.555118
## 2003              3 -1.7774485 7.777449

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,1,0)
## Q* = 7.0945, df = 10, p-value = 0.7165
## 
## Model df: 0.   Total lags used: 10

This backtest model is only computing the value 3 as a prediction. We want the prediction within 1 point (1000 basis points) of the original Fed Model object Fed_Fund_Rate. After viewing the results from check residuals, we noticed that the data didn’t check all the boxes referenced above and P-Value is .71 (71%) which gives us a confidence interval of 29%. Let’s manually select the ARIMA.

Backtest_Fed_Model_Forecast_Update

##      Point Forecast    Lo 30    Hi 30
## 1993       3.659781 3.048595 4.270966
## 1994       4.577001 3.706645 5.447357
## 1995       5.032248 4.032319 6.032176
## 1996       4.873646 3.820492 5.926801
## 1997       5.353607 4.263323 6.443891
## 1998       5.021209 3.864710 6.177708
## 1999       4.502486 3.305557 5.699416
## 2000       4.312817 3.059936 5.565699
## 2001       4.537356 3.203546 5.871165
## 2002       4.279734 2.863492 5.695976
## 2003       4.452451 2.981904 5.922997

checkresiduals(Backtest_Fed_Model_Forecast_Update)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(5,1,6)
## Q* = 6.76, df = 3, p-value = 0.07995
## 
## Model df: 11.   Total lags used: 14

Let’s update our confidence interval to 92% based on our new p value.

Backtest_Fed_Model_Forecast_Update = forecast(
  Fed_Model_1993_2003, level = c(92), h = 11)

Verify if the model works by using true/false rubric with basis point tolerance .³

Year	Original_Fed_Rate	Model_Fed_Fund_Rate	Fed_Diff	Within_Tolerance
Federal Reserve's Fed Fund Rate Model
HardvardX Capstone Project 2022
1993	3.00	3.7	0.70	TRUE
1994	5.50	4.6	-0.90	TRUE
1995	5.50	5.0	-0.50	TRUE
1996	5.25	4.9	-0.35	TRUE
1997	5.50	5.4	-0.10	TRUE
1998	4.75	5.0	0.25	TRUE
1999	5.50	4.5	-1.00	TRUE
2000	6.50	4.3	-2.20	TRUE
2001	1.75	4.5	2.75	FALSE
2002	1.25	4.3	3.05	FALSE
2003	1.00	4.5	3.50	FALSE
Portions of this data is from the Reference Section
Fed Funds Rate is based on data from 1993-2003

Create chart with Backtest Model Federal Funds Rate vs Actual Federal Funds Rate (1993-2003).

As we can see the majority of the rates are within our model. In the Year 2000, the U.S. Economy experienced the Dot.com crash. That recessionary event caused the Federal Reserve to cut rates which explains why rates crashed downward.

10.4 Create a Future Federal Funds Rate YoY Forecasting Model

## 
##  Box-Ljung test
## 
## data:  Fed_Model_Forecast
## X-squared = 3.4475, df = 1, p-value = 0.06335

## 
##  Box-Ljung test
## 
## data:  Fed_Model_Forecast
## X-squared = 11.303, df = 5, p-value = 0.0457

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(5,1,6)
## Q* = 13.158, df = 3, p-value = 0.004308
## 
## Model df: 11.   Total lags used: 14

11. Results:

All of our goals where achieved:

We identified the best correlation was a positive correlation with the Federal Funds Rate and Inflation Rate YoY:
- correlation is 0.7710506
- R² of 88% this means that 12% of variance is explained by outliers deriving from geopolitical and economic factors instead of unknown factors.
For our Inflation Rate YoY Machine Learning Model:
- We identified the best Model for backtesting had an ARIMA (6,2,1) with a 87% confidence interval.
- We identified the best Model for forecasting had an ARIMA (6,2,1) with a 90% confidence interval.
- Backtest Code & Forecast Code:

Inflation_Model_1993_2003 = arima(
  fixed = NULL, Inflation_Model_1993_Time, order = c(6,2,1), 
  transform.pars=TRUE)
Backtest_Inflation_Model_Forecast_1 = forecast(
  Inflation_Model_1993_2003, level = c(87), h = 11)

    True_Inflation_Model = arima(
      fixed = NULL, Inflation_Model_Time, order = c(6,2,1), 
      transform.pars=TRUE)
Inflation_Model_Forecast_Update = forecast(
  True_Inflation_Model, level = c(90), h = 11)

For our Federal Reserve Federal Funds Rate Model:
- We identified the best Model for backtesting had an ARIMA (5,1,6) with a 92% confidence interval.
- We identified the best Model for forecasting had an ARIMA (5,1,6) with a 95% confidence interval.
- Backtest Code & Forecast Code:

Fed_Model_1993_2003 = arima(
  fixed = NULL, Fed_Model_1993_Time, order = c(5,1,6),  
  transform.pars=TRUE)
Backtest_Fed_Model_Forecast_Update = forecast(
  Fed_Model_1993_2003, level = c(92), h = 11)

Fed_Model_True = arima(
  fixed = NULL, transform.pars=TRUE, Fed_Model_Time, order = c(5,1,6))
Fed_Model_Forecast = forecast(Fed_Model_True, level = c(95), h = 11)

12. Conclusion:

Utilizing Pearson’s Correlation Coefficient (r) tool can show how different variables may have zero, inverse or positive correlation. In this case, we know that economic and geopolitical factors can drive outliers affecting Pearson’s Correlation Coefficient (r) calculations by skewing the calculated results.

Geopolitical and economic outliers can also affect monetary policy, which the Federal Reserve drives. We can also conclude that the Federal Funds Rate and the Inflation Rate YoY have a positive but moderate correlation between 1929-1950 but a significantly positive correlation from 1951-2017 once the Federal Reserve utilized all the tools required to help fight inflation.

Lastly, adding all these factors into our Machine Learning model proved essential in backtesting and predicting future inflation rates YoY and Fed funds rates.

Note: required data sets are embedded for simplicity↩︎
Correction: row 3 (1995) should be FALSE↩︎
Correction: Row 8 (2000) should be FALSE.↩︎

Inflation: Finding the Correlations to Build a Machine Learning Model for Backtesting and Forecasting

Arthur Richardson

9/5/2022